Endowment Insurance as Term Plus Pure Endowment

Why an Endowment Has Two Components

A continuous endowment insurance contract makes a payment in either of two circumstances: a death benefit if the insured dies during the term, or a maturity benefit if the insured survives to the end. The payments occur at different times and depend on different outcomes. Separating them makes the contract’s expected present value easier to understand and calculate.

This tutorial considers a life aged $x$, a term of $n$ years, and a constant effective annual interest rate $i$. Write $v=1/(1+i)$ and $\delta=\ln(1+i)$, so the discount factor for a payment at time $t$ is $v^t=e^{-\delta t}$. Unless stated otherwise, the death benefit is payable immediately at death and the maturity benefit is payable at time $n$ if the insured is then alive.

Definition: A continuous endowment insurance with death benefit $B$ and maturity benefit $M$ pays $B$ at the moment of death if death occurs before the end of the $n$-year term, and pays $M$ at time $n$ if the insured survives the term. The two benefits may be equal, but need not be.

Let $T_x$ denote the insured’s future lifetime. The death-payment event is $T_x\le n$; the maturity-payment event is $T_x>n$. These events are mutually exclusive and, together, cover the possible outcomes. In a continuous lifetime model, the distinction between $T_x<n$ and $T_x\le n$ does not affect the expected present value when there is no probability mass at exactly $n$.

The Decomposition

The unit death-benefit component is continuous term insurance: it pays one at death if death occurs within the term and nothing otherwise. The unit maturity component is a pure endowment: it pays one at time $n$ only if the insured survives to that time. For this tutorial, denote their expected present values by $\bar{A}^{1}_{x:\overline{n}|}$ and ${}_nE_x$, respectively. In particular, ${}_nE_x=v^n{}_np_x$.

Theorem: For a continuous endowment with death benefit $B$ and maturity benefit $M$, the expected present value is $$ B\bar{A}^{1}_{x:\overline{n}|}+M\,{}_nE_x = B\bar{A}^{1}_{x:\overline{n}|}+M v^n{}_np_x. $$ When the two benefits are the same amount $M$, its expected present value is $M\bigl(\bar{A}^{1}_{x:\overline{n}|}+{}_nE_x\bigr)$.

Here ${}_np_x$ is the probability that a life aged $x$ survives $n$ years. The formula assumes the stated benefit amounts, payment times, survival probabilities, and interest basis; it is an expected-present-value result, not a statement that the actual contract pays both benefits. If the maturity payment is not made on survival, or the death benefit is not paid for death during the term, the contract is different and the decomposition must be adjusted.

Why the Formula Holds

The decomposition follows directly by expressing the contract’s present value as a random variable. Let $Z$ be the present value of the benefits paid. A death before maturity produces a discounted death benefit; survival to maturity produces a discounted maturity benefit. Thus

Z = Bv^{T_x}\mathbf{1}_{\{T_x\le n\}} + Mv^n\mathbf{1}_{\{T_x>n\}}.

Taking expectations and using the linearity of expectation gives the two separate expected present values. The first term is $B$ times the expected present value of unit continuous term insurance. For the second term, $v^n$ is fixed, and the indicator is one exactly when the life survives $n$ years. Its expectation is therefore the survival probability ${}_np_x$.

\begin{aligned} E[Z] &=B E\!\left[v^{T_x}\mathbf{1}_{\{T_x\le n\}}\right] +M v^n E\!\left[\mathbf{1}_{\{T_x>n\}}\right]\\ &=B\bar{A}^{1}_{x:\overline{n}|}+M v^n{}_np_x\\ &=B\bar{A}^{1}_{x:\overline{n}|}+M\,{}_nE_x. \end{aligned}

This argument also explains why the two components are added even though the contract pays only one benefit in any one outcome: expectation is linear, and each component is zero in the outcome where it does not apply. The formula does not require the death and maturity amounts to be equal.

Calculating the Components Under Constant Force

For numerical work, suppose the force of mortality is constant at $\mu$ throughout the term and the force of interest is constant at $\delta$. Then ${}_tp_x=e^{-\mu t}$, so the pure endowment value is $e^{-(\mu+\delta)n}$. The continuous term component is the expected discounted payment at death during the term. Under this model, the density of death at time $t$ is $\mu e^{-\mu t}$, and discounting that payment by $e^{-\delta t}$ gives

\bar{A}^{1}_{x:\overline{n}|} =\int_0^n e^{-\delta t}\mu e^{-\mu t}\,dt =\frac{\mu}{\mu+\delta}\left(1-e^{-(\mu+\delta)n}\right).

The formula requires $\mu+\delta>0$ in its displayed quotient form. If $\mu+\delta=0$, the integral itself gives the appropriate limiting value. The constant-force expression is not appropriate if mortality or the force of interest changes over the term; in that case, use the applicable term-insurance and pure-endowment values on the stated assumptions.

Worked Example: Equal Death and Maturity Benefits

A hypothetical contract covers a life aged $x$ for 8 years. It pays $120,000 immediately at death during the term and $120,000 at year 8 if the insured survives. Assume a constant force of mortality $\mu=0.05$ and an effective annual interest rate $i=0.0304545$, corresponding to $\delta=0.0300$. Find the expected present value. Round money to the nearest cent.

Method. Calculate the unit continuous term component and unit pure endowment separately, then multiply each by its applicable benefit. The death and maturity payments are alternative outcomes, not simultaneous payments.

First, $\mu+\delta=0.05+0.03=0.08$, and $(\mu+\delta)n=0.08(8)=0.64$. Using $e^{-0.64}=0.527292424$, the unit term-insurance value is

\bar{A}^{1}_{x:\overline{8}|} =\frac{0.05}{0.08}(1-e^{-0.64}) =0.625(1-0.527292424) =0.625(0.472707576) =0.295442235.

The unit pure-endowment value is

{}_8E_x=e^{-0.64}=0.527292424.

The death-benefit contribution is $120{,}000(0.295442235)=\$35{,}453.07$, and the maturity-benefit contribution is $120{,}000(0.527292424)=\$63{,}275.09$. Therefore,

\operatorname{EPV} =\$35{,}453.07+\$63{,}275.09 =\$98{,}728.16.

Check. The unit values sum to $0.295442235+0.527292424=0.822734659$, so the total is $120{,}000(0.822734659)=\$98{,}728.16$ to the nearest cent. The two calculations agree.

Different Death and Maturity Amounts

When the amounts differ, apply each benefit only to its own component. The death-benefit amount multiplies the term-insurance value, while the maturity amount multiplies the pure-endowment value. Using a single amount for both components would change the contract being valued.

Worked Example: Unequal Benefits and a Survival Payment

A hypothetical 5-year contract pays $100,000 immediately on death during the term and $60,000 at year 5 on survival. Assume a constant force of mortality $\mu=0.04$ and an effective annual interest rate $i=0.06184$, corresponding to $\delta=0.0600$. Calculate the expected present value, rounded to the nearest cent.

Method. Use the same mortality and interest assumptions to find both unit values, but scale them by the separate contract amounts. Here $\mu+\delta=0.10$ and $(\mu+\delta)n=0.50$, with $e^{-0.50}=0.606530660$.

\bar{A}^{1}_{x:\overline{5}|} =\frac{0.04}{0.10}(1-e^{-0.50}) =0.4(1-0.606530660) =0.4(0.393469340) =0.157387736.

Thus the death-benefit contribution is

\$100{,}000(0.157387736)=\$15{,}738.77.

The pure-endowment value is $e^{-0.50}=0.606530660$, so the maturity-benefit contribution is

\$60{,}000(0.606530660)=\$36{,}391.84.

Adding the contributions gives an expected present value of $\$15{,}738.77+\$36{,}391.84=\$52{,}130.61$. This separates the value attributable to death cover from the value attributable to survival to year 5.

Check. The unit components are each nonnegative, and the maturity amount is smaller than the death amount. The maturity contribution is larger here because its unit value is larger; the calculation uses the actual amount attached to each event.

Worked Example: Recovering a Benefit From the Expected Present Value

A hypothetical 4-year endowment pays the same amount $M$ on death during the term or survival to maturity. Assume a constant force of mortality $\mu=0.10$ and an effective annual interest rate $i=0.05127$, corresponding to $\delta=0.0500$. The expected present value is reported as $67,968.31. Find $M$ to the nearest dollar.

Method. Since the benefit is the same in both cases, first add the two unit values. Here $\mu+\delta=0.15$, $(\mu+\delta)n=0.60$, and $e^{-0.60}=0.548811636$.

\bar{A}^{1}_{x:\overline{4}|} =\frac{0.10}{0.15}(1-e^{-0.60}) =\frac{2}{3}(0.451188364) =0.300792243, $$ $$ {}_4E_x=e^{-0.60}=0.548811636.

The unit endowment value is therefore $0.300792243+0.548811636=0.849603879$. Since the expected present value is $M$ times this amount,

M=\frac{\$67{,}968.31}{0.849603879} \approx \$80{,}000.00.

To the nearest dollar, $M=\$80{,}000$. Check. Multiplying the unrounded unit value by $80,000 gives approximately $67,968.3103, which rounds to the stated $67,968.31.

Common Mistakes and Why the Distinction Matters

The decomposition is useful because it identifies which part of an endowment value is due to death cover and which part is due to the guaranteed payment on survival. It also makes changes to the contract easier to evaluate: changing only the maturity amount affects only the pure-endowment component, while changing only the death benefit affects only the term-insurance component.

Applying the death benefit at maturity. The term component covers death during the term; the maturity component applies only if the insured survives. Do not include the maturity benefit in the death-payment amount unless the contract explicitly specifies that design.
Discounting the maturity payment to death time. A maturity benefit is paid at time $n$, so its unit value is $v^n{}_np_x$, not a discount factor based on a possible time of death.
Using the wrong amount for one component. If $B\ne M$, use $B\bar{A}^{1}_{x:\overline{n}|}+M{}_nE_x$. Multiplying the sum of unit values by $B$ incorrectly assumes the maturity benefit is also $B$.
Confusing the maturity amount with its expected present value. The amount paid on survival is $M$; its expected present value is $M v^n{}_np_x$, which accounts for both survival and discounting.

For example, if a hypothetical calculation with the unequal benefits in the previous example incorrectly used $100,000 for both components, it would give $100{,}000(0.157387736+0.606530660)=\$76{,}391.84$, rather than $52,130.61. The correction is to multiply the pure-endowment value by the actual $60,000 maturity amount: $\$15{,}738.77+\$36{,}391.84=\$52{,}130.61$.

Key takeaway: A continuous endowment is the sum of two present-value components: term insurance for death during the term and a pure endowment for survival to maturity. Scale each component by its own benefit amount, and use the contract’s actual timing and assumptions.

Check Your Understanding

For each problem, identify the death and maturity components before combining them. Round dollar answers to the nearest cent unless another rounding instruction is given.

A hypothetical 10-year contract pays $1 on death during the term and $1 on survival to year 10. Under constant forces $\mu=0.03$ and $\delta=0.02$, calculate its expected present value. (Answer: $0.8426$)
A 4-year contract pays $50,000 immediately on death and $100,000 at maturity if the insured survives. Assume constant forces $\mu=0.05$ and $\delta=0.05$. Find its expected present value. (Answer: $75,274.00)
A hypothetical 6-year contract pays $40,000 on death during the term and $20,000 on survival to maturity. Assume constant forces $\mu=0.02$ and $\delta=0.03$. Calculate the expected present value. (Answer: $18,963.27)
A 3-year endowment pays the same amount $M$ on death during the term or survival to maturity. Assume constant forces $\mu=0.04$ and $\delta=0.06$. If its expected present value is $42,224.55, find $M$ to the nearest dollar. (Answer: $50,000)
A hypothetical 12-year contract has a $25,000 maturity benefit and an unspecified death benefit. Under constant forces $\mu=0.03$ and $\delta=0.05$, find the expected present value of the maturity component alone. (Answer: $9,572.32)

What You'll Learn